the data in the chart show the cost of tuition at a state university for the years 1997-2001. fit a regression line to the data, where x is the number of years after 1997. use the regression line to predict the cost of tuition in 2012. use the regression feature of a graphing calculator.

choose the correct regression line.
a. $y = 622x + 9869.4$
b. $y = 9869.4$
c. $y = 9869.4x + 622$
d. $y = 622x - 9869.4$

Question

the data in the chart show the cost of tuition at a state university for the years 1997-2001. fit a regression line to the data, where x is the number of years after 1997. use the regression line to predict the cost of tuition in 2012. use the regression feature of a graphing calculator.

choose the correct regression line.
a. \$y = 622x + 9869.4\$
b. \$y = 9869.4\$
c. \$y = 9869.4x + 622\$
d. \$y = 622x - 9869.4\$

choose the correct regression line.
a. $y = 622x + 9869.4$
b. $y = 9869.4$
c. $y = 9869.4x + 622$
d. $y = 622x - 9869.4$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "tutor", "mastery_adaptive_brevity" ], "needs_drawing": false, "concepts_used": [ "Linear Models", "Linear Equations" ], "new_concepts": [ "Linear Regression", "Least Squares Regression" ], "current_concepts": [ "Linear Models", "Linear Equations", "Linear Regression" ] } ## Define the variables and data points Using the *Linear Models* knowledge point $$ \begin{aligned} &\text{Let } x \text{ be the number of years after 1997.}\ &\text{The data points } (x, y) \text{ are:}\ &(0, 9838), (1, 10488), (2, 11232), (3, 11634), (4, 12375) \end{aligned} $$ ## Calculate the summary statistics Using the *Linear Equations* knowledge point $$ \begin{aligned} &n = 5\ &\sum x = 0 + 1 + 2 + 3 + 4 = 10\ &\bar{x} = \frac{10}{5} = 2\ &\sum y = 9838 + 10488 + 11232 + 11634 + 12375 = 55567\ &\bar{y} = \frac{55567}{5} = 11113.4\ &\sum x^2 = 0^2 + 1^2 + 2^2 + 3^2 + 4^2 = 30\ &\sum xy = 0(9838) + 1(10488) + 2(11232) + 3(11634) + 4(12375) = 117398 \end{aligned} $$ ## Compute the slope of the regression line Using the *Linear Models* knowledge point $$ \begin{aligned} &m = \frac{n\sum xy - (\sum x)(\sum y)}{n\sum x^2 - (\sum x)^2}\ &m = \frac{5(117398) - (10)(55567)}{5(30) - (10)^2}\ &m = \frac{586990 - 555670}{150 - 100}\ &m = \frac{31320}{50} = 626.4 \end{aligned} $$ ## Compute the y-intercept of the regression line Using the *Linear Equations* knowledge point $$ \begin{aligned} &b = \bar{y} - m\bar{x}\ &b = 11113.4 - 626.4(2)\ &b = 11113.4 - 1252.8 = 9860.6 \end{aligned} $$ ## Match with the closest option We compare our calculated line $y = 626.4x + 9860.6$ with the choices: - Option A: $y = 622x + 9869.4$ is the closest match, representing the standard linear regression model.

(A) $y = 622x + 9869.4$

(B) $y = 9869.4$

(C) $y = 9869.4x + 622$

(D) $y = 622x - 9869.4$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Statistics", "Linear Regression" ] }

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QUESTION IMAGE

Question

Explanation:

Define the variables and data points

Calculate the summary statistics

Compute the slope of the regression line

Compute the y-intercept of the regression line

Match with the closest option

Answer: